Problem 48 - Elementary Algebra

Question

For the following problems, use the zero-factor property to solve the equations. $$ (x+5)(x+4)=0 $$

Step-by-Step Solution

Answer

Answer: The solutions for x are -5 and -4.

Step 1: Identify the factors

In this equation, the factors are (x+5) and (x+4).

Step 2: Apply the zero-factor property

To apply the zero-factor property, set each factor equal to zero: (x+5)=0 and (x+4)=0

Step 3: Solve for x in each equation

Solve for x in the first equation: x+5=0 x=-5 Solve for x in the second equation: x+4=0 x=-4

Step 4: Write the final solution

The solutions for x are -5 and -4. This can be expressed as: x = {-5, -4}

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a core skill in algebra. A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\). Quadratic equations can have two solutions because they are degree 2 polynomials. There are several methods to solve these equations, including:

• Factoring
• Using the quadratic formula
• Completing the square

One of the most straightforward methods for equations that are factorable is the Zero-Factor Property. This method simplifies the problem by splitting the equation into factors that can be easily solved. By breaking down the equation into its components, you create simpler equations that can be tackled individually. Then, you solve these smaller equations to find the roots of the original equation.

Factoring

Factoring is a powerful tool used to simplify polynomial expressions. When solving quadratic equations by factoring, we aim to express the quadratic equation in a product of two binomials. Consider a quadratic equation written as \((x + m)(x + n) = 0\). The task is to find two numbers, \(m\) and \(n\), such that when multiplied they give \(c\), and when added, they give \(b\).
For example, the expression \((x+5)(x+4)=0\) is already factored, with \(m=5\) and \(n=4\). This method allows us to use the Zero-Factor Property to find the variable's value by setting each factor to zero. When factors are correctly identified, the quadratic equation can be split into linear equations that are much easier to solve.

Roots of Equations

Roots, or solutions, of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). In simpler terms, the roots are the points where the graph of the equation intersects the x-axis. For a factored quadratic equation, such as \((x+5)(x+4)=0\), the roots can be easily determined using the Zero-Factor Property.
The Zero-Factor Property states that if a product of two terms is zero, then at least one of the terms must be zero. Therefore, set each factor equal to zero and solve for \(x\).

• For \(x+5=0\), subtract 5 from both sides to get \(x=-5\).
• For \(x+4=0\), subtract 4 from both sides, resulting in \(x=-4\).

Thus, the solution or the roots of the equation \((x+5)(x+4)=0\) are \(x=-5\) and \(x=-4\). Recognizing the roots is crucial because it allows us to understand the solutions in the context of the equation.